Global Optimization of the Difference of two Increasing Plus-Convex-Along-Rays Functions
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
GLOBAL OPTIMIZATION OF THE DIFFERENCE OF TWO INCREASING PLUS-CONVEX-ALONG-RAYS FUNCTIONS∗ H. SHAHRIARIPOUR Department of Mathematics, Graduate University of Advanced Technology, Kerman, Iran E-mail : [email protected]
H. MOHEBI† Department of Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, P.O. Box: 76169133, Postal Code: 7616914111, Kerman, Iran E-mail : [email protected] Abstract The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such as support sets and subdifferentials. Finally, as an application, we characterize the maximal elements of the support set of strictly IPCAR functions and give optimality conditions for the global minimum of the difference between two IPCAR functions. Key words
increasing plus-convex-along-rays function; support set; maximal element; global minimum; DC-function
2010 MR Subject Classification
1
90C46; 26A48; 26A51; 26B25
Introduction
The theory of abstract convexity, also called convexity without linearity, is a powerful tool that allows us to extend many facts from classical convex analysis to more general frameworks. It has been the focus of active research for the last fifty years because of its many applications in functional analysis, approximation theory, and nonconvex analysis. Nevertheless, just like convex analysis, the development of abstract convexity has been mainly motivated by applications to optimization. The works [2, 5–8, 13, 16, 17, 22, 27, 31–33, 38, 41] use abstract convexity for applications to nonconvex optimization. A deep study on abstract convexity can be found in the seminal book of Alex Rubinov [34]; see also the monograph of Ivan Singer [40]. Abstract convexity reflects one of the fundamental concepts of convex analysis, which is the fact that ∗ Received
June 6, 2019; revised July 17, 2020. The second author was partially supported by the Mahani Mathematical Research Center, Iran, grant no: 97/3267. † Corresponding author: H. MOHEBI.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
every lower semicontinuous convex function f is the upper envelope of affine functions [24, 42]. More precisely, at every point x, we have f (x) = sup{h(x) : h is an affine function, h ≤ f }.
(1.1)
Most results in convex analysis are consequences of two important aspects of (1.1): (i) the “supremum” operation, and (ii) the set over which this supremum is taken. Results that depend on aspect (ii) are likely to depend on specific properties of linear/affine functions. How does one distinguish which facts from convex analysis follow from the “upper envelope“ operation, and which follow
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